Generally fuzzy logic systems utilize rules against which inputs are evaluated in order to formulate an output. In the present specification, a rule refers to a fuzzy proposition, which is indicated as A→B, where A is the rule input and B is the rule output. For example, in the phrase “red cars are liked”, the rule input is “red cars” and the rule output is “liked”. The input is a fuzzy set that may or may not be identical to the rule input. For example, “green cars” and “orange vans” would be inputs. The output is a conclusion inferred by applying the rule to the input. The conclusion may or may not be the same as the rule output depending on the input. A rule excludes certain outputs absolutely because it is the result of many observations that lead to a firm conclusion that nothing other than B will occur if A is true. An “example” is defined as “a single observation of B together with A”. If situation A recurs, outputs other than B are deemed possible
Existing fuzzy logic systems have limited decision making capabilities and therefore are less likely to emulate a desired system requiring reasoning that is similar to informal human reasoning. These limitations may be described as follows:
1) Existing fuzzy logic implication operators do not generate outputs corresponding to intuitive ideas for the output if the input does not match the rule input exactly.
For example, in the case of mismatch between input and rule input, informal logic postulates for the output an envelope of possibility should spread around the rule output, and spread wider as the input becomes less similar to the rule input. This spreading reflects increased uncertainty about the range of possible outputs. If the input is “sort of” like the rule input, the output should be “sort of” like the rule output, where “sort of” means an increased degree of fuzziness and/or a wider support set.
One expects outputs closer to the rule output to be more possible than remote outputs. For example, if a vehicle is “orange car”, one does not expect “intensely disliked” (an output remote from the rule output “liked”) to be just as possible as “somewhat liked” (an output close to the rule output “liked”).
Existing fuzzy logic generates basically two types of outputs if the input and rule input do not match exactly, exemplified by a Zadeh implication and a Sugeno implication. In the former, the envelope of possibility has a core identical to the rule output and infinite flat tails whose height is proportional to the mismatch. In the latter, the envelope of possibility does not spread at all but becomes increasingly subnormal as the mismatch increases.
2) Existing fuzzy logic requires a complete set of overlapping rules covering all possible combinations of inputs, whereas human beings can reason from a very sparse set of rules or examples.
A complete set of overlapping rules is required for fizzy logic because only logical operations (as opposed to arithmetical operations) are applied to the inputs to get the output, and logical operations can only be applied to fuzzy sets that intersect to some degree. Existing fuzzy logic can not function with disjoint sets of rules, whereas human beings can function by filling in the blank spaces in a rule input “grid”. For example, if you knew “red cars are liked” and “white cars are hated”, you would guess that “pink cars elicit indifference”. Humans do not need a new rule for this situation.
When using the newly created rules, human beings assume that the output is fuzzier than it would be if the input matched the rule input exactly. This increasing fuzziness corresponds to the desired envelope of possibility described in 1). For example, your conclusion about pink cars would not be very certain because you have definite information only about red and white cars. You therefore hedge your conclusion with words to make the conclusion fuzzier and to indicate doubt about the conclusion: “Most likely people are indifferent to pink cars, but it's also somewhat possible they might hate them or love them, I can't be sure”
Expert knowledge is currently formulated in fuzzy logic as a complete set of rules. However, in much of informal reasoning, expert knowledge is represented by: a sparse set of examples or rules, knowledge of how to deviate from those rules, and a measure of how far to trust those deviations, all of which is not represented by existing fuzzy logic.
3) Existing Fuzzy Logic Does not Smoothly Bridge the Gap Between Examples and Rules.
In current practice, a large number of discrete data points (examples) are sampled, clustering analysis or the application of a neural net follows, and then a complete fizzy rule set is extracted. A human being, on the other hand, will start reasoning from one example, correct his reasoning on getting a second example, and with no switchover from one mathematical approach to another, continue formulating new rules from however many examples as are available.
4) Existing Fuzzy Logic Does not Explicitly Encode Degrees of Continuity and Chaos.
Human beings assess certain environments as more chaotic than others. In chaotic environments, a small change in the input could lead equally well to a large change in the output or to a small change. In environments where continuity prevails, a small change in the input leads to a change in the output roughly proportional to the change in input, but the proportionality constant is only vaguely known, or only a vague upper limit on its absolute magnitude is known.
For example, suppose that the temperature in a certain city is about 20° C. and a person wishes to know what the temperature is in another city that is 300 kn away. In general, temperature is a continuous function of latitude and longitude, however, if there are mountain ranges, elevation differences, or large bodies of water, discontinuity is possible.
If the person thinks that this particular terrain is flat and without bodies of water, he/she would make the assumption of continuity; and the envelope of possible temperatures will be a fuzzy number centered around 20° C. Experience says that temperatures change at most one or two degrees for every hundred kilometers, therefore, a person would know approximately how far the envelope of possible temperatures would spread outside the original number “about 20 C”.
If the two cities are at different elevations, then the estimate envelope for the second city may no longer symmetrical around the fizzy number “about 20C”. Five degrees is just as possible as fifteen degrees, which should be represented by the fuzzy logic system.
5) In Existing Fuzzy Measure Theory, the Concepts of Belief and Plausibility have been Applied only to Assertions.
Expert opinion and evidence currently consist of assertions, not rules. Assertions are statements of fact such as “This car is red”. People however apply these belief and plausibility concepts to new rules entailed from established rules. For example, if the rule “red cars are liked” is true, and there is no other information, then “blue cars are liked” is 100% plausible, since there is no evidence, in the form of a rule about blue cars, that would contradict the entailed proposition “blue cars are liked”. However, neither is there evidence to support the entailed proposition “blue cars are liked”, hence that proposition is believable to degree zero.
Any conclusions drawn from entailed rules should inherit these degrees of belief and plausibility derived from the entailment before they can be used for decision making.
6) Many systems to which fuzzy expert systems are applied have some fractal geometry. Existing fizzy logic expert systems do not explicitly incorporate the ability to adequately simulate such systems.
There is therefore a need for a fuzzy logic system that mitigates at least some of the disadvantages of existing systems while achieving some of the advantages as described above.
This invention seeks to provide a solution to the problem in fuzzy logic systems wherein user rule input does not match a rule exactly. Accordingly this invention provides for bridging the gap between non-matching rules and rule inputs by creating envelopes of possibility for an output, the output having different shapes and rates of spreading and wherein the rate of spreading is a function of distance between the user input and the rule input. The desired shape of the envelope of possibility is a system parameter determined at set tip by an expert, while the similarity between the user input and the rule input may be measured by existing measures or by a novel measure. The rate of spreading of the envelope as a function of the dissimilarity between the input and the rule input is determined by the expert. It may also depend on the location of the input in input space or other parameters of the input and the rule input.
For multidimensional inputs, that is inputs where more than one attribute is defined for each input, the different dimensions may be weighted differently when calculating the distance between the multidimensional input and the multidimensional rule input, to reflect greater sensitivity of the output to some of the dimensions of the input. A weight function also makes it possible for one input dimension to “compensate” for another in the generally accepted sense of the word.
This invention further provides a method to eliminate the requirement for a complete set of overlapping rules. Instead, it is possible to calculate degrees of similarity between disjoint fuzzy sets using a distance function in order to interpolate or extrapolate from sparse examples or rules. Fuzzy limits can be set on the vaguely known possible rate of change of the output and it is possible to reconcile contradictory inputs, and choose the appropriate pattern to interpolate or extrapolate from.
This invention further seeks to make it possible for fizzy logic to smoothly bridge the gap between examples and rules. By providing means to calculate degrees of similarity (or distance) between two fuzzy sets, between two point data examples, between a fuzzy number and a point data example, or between two fuzzy numbers, it is possible to bridge the gap between examples and rules. Existing measures of set intersection or similarity may also be used but for existing measures, interpolation/extrapolation cannot be done if the input does not intersect a rule input
This invention also seeks to make it possible to encode the degree to which chaos or continuity occurs. A new family of fuzzy implications, of which the Zadeh implication is a special case, makes it possible. The degree of chaos or continuity may depend on the location of the input in input space. An output can be continuous in one of the input dimensions but chaotic in another if the inputs are multidimensional.
This invention seeks to provide a solution for the problem where the concepts of belief and plausibility are only applied to assertions, not to propositions.
Using the kernel of the new fuzzy implication operator, one can arrive at a degree of plausibility. an entailed proposition, and an envelope of possible conclusions for a given input.
Using set intersection Or other distance measures, the strength of the chain of evidence and reasoning linking the data to the conclusion can be calculated and thus obtain an envelope of belief. The difference between the envelopes of belief and possibility measures all the vagueness, uncertainty gaps, contradiction, and probabilistic nature of the rules and the input data as well as the mismatch between the inputs and the rule inputs. The degree to which an assertion is proven and the degree to which it is merely possible can be quantified.
This invention seeks to provide a method for malting use of the fractional dimension or other parameters of fracial systems that current filmy systems do not make use of to calculate an envelope of possibility for fractal systems.
Using the new fuzzy implication operator with the appropriate kernel and the appropriate new distance measure, the envelope of possibility can be found for a system characterized by a vaguely specified fractal dimension.
In accordance with this invention there is provided in an expert system a method for determining an outcome from a set of inputs, the method comprising the steps of determining: a set of parameters by an expert establishing at least one rule using at least two sets of parameters as input and output; according values to each of a selected ones of sets of parameters; computing an envelope of possibility by operating on inputs and selected ones of said sets of parameters (a spreading function or kernel for the implication operator, curve fitting procedure for interpolation/extrapolation, distance functions, weights and weight function); computing a belief envelope; comparing possibility and belief envelopes with predetermined criteria to determine the envelope of possibility is sufficiently narrow; if the system is being used for assessing evidence supporting an assertion, compare possibility and belief envelopes to assertion in question; output based on envelope of possibility must be selected if the system is being used for assessing evidence, either advise user to collect more input data to confirm/disconfirm assertion to the required degree or select output.